arXiv subjects

Martin Bauer

Publications and source records attributed to Martin Bauer.

At least 37 records · Page 2Linked to original sources

arXiv2025

Path constrained unbalanced optimal transport

Dynamical formulations of optimal transport (OT) frame the task of comparing distributions as a variational problem which searches for a path between distributions minimizing a kinetic energy functional. In applications, it is frequently natural to require paths of distributions to satisfy additional conditions. Inspired by this, we introduce a model for dynamical OT which incorporates constraints on the space of admissible paths into the framework of unbalanced OT, where the source and target measures are allowed to have a different total mass. Our main results establish, for several general families of constraints, the existence of solutions to the variational problem which defines this path constrained unbalanced optimal transport framework. These results are primarily concerned with distributions defined on a Euclidean space, but we extend them to distributions defined over parallelizable Riemannian manifolds as well. We also consider metric properties of our framework, showing that, for certain types of constraints, our model defines a metric on the relevant space of distributions. This metric is shown to arise as a geodesic distance of a Riemannian metric, obtained through an analogue of Otto's submersion in the classical OT setting.

math.OC↗

arXiv2025

Breakdown of smooth solutions to the subcritical EPDiff equation

We consider the EPDiff equation on $\mathbb{R}^n$ with the integer-order homogeneous Sobolev inertia operator $A=(-Δ)^k$. We prove that for arbitrary radial initial data and a sign condition on the initial momentum, the corresponding radial velocity solution has $C^1$ norm that blows up in finite time whenever $0\le k<n/2+1.$ Our approach is to use Lagrangian coordinates to formulate EPDiff as an ODE on a Banach space, enabling us to use a comparison estimate with the Liouville equation. Along the way we derive the Green function in terms of hypergeometric functions and discuss their properties. This is a step toward proving the general conjecture that the EPDiff equation is globally well-posed for any Sobolev inertia operator of any real order $k$ if and only if $k\ge n/2+1$.

math.AP↗

arXiv2025

Sobolev Metrics on Spaces of Discrete Regular Curves

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and considers the induced Riemannian metric on a finite dimensional approximation space. Surprisingly, the theoretical properties of the corresponding finite dimensional Riemannian manifolds have not yet been studied in detail, which is the content of the present article. Our main theorem concerns metric and geodesic completeness and mirrors the results of the infinite dimensional setting as obtained by Bruveris, Michor and Mumford.

math.DG↗

arXiv2025

On the Validity of Bounds on Light Axions for $f\lesssim10^{13}$ GeV

Light bosonic dark matter fields that can be treated like a classical wave have non-linear field values close to massive bodies. Here we make the important observation that the quadratic interactions of axion dark matter lead to non-perturbative axion field values for values of the decay constant of $f\lesssim 10^{13}$ GeV and masses $m_a\lesssim 7\times10^{8}/f\ {\rm eV^2}$ and generalise this result for axion-like particles. We identify experimental observables impacted by this effect.

hep-ph↗

arXiv2025

Regularity and completeness of half-Lie groups

Half Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth, but left translations are merely required to be continuous. The main examples are groups of $H^s$ or $C^k$ diffeomorphisms and semidirect products of a Lie group with kernel an infinite dimensional representation space. Here, we investigate mainly Banach half-Lie groups, the groups of their $C^k$-elements, extensions, and right invariant strong Riemannian metrics on them: surprisingly the full Hopf--Rinow theorem holds, which is not the case in general even for Hilbert manifolds.

math.DG↗

arXiv2024

Generalised hydrogen interactions with $\texttt{CINCO}$: a window to new physics

We present semi-analytic solutions for atomic transition rates in hydrogenic atoms induced by scalar, pseudoscalar, vector, axial-vector, and tensor interactions. Our results agree with quantum electrodynamics predictions to $\sim 0.005\,\%$ precision, and further allow us to calculate absorption and emission rates for axions, hidden photons, light scalars or other dark matter candidates for hydrogen and hydrogenic ions. These results can be used to inform searches for light new physics as well as in calculations relevant to searches for fifth forces or varying fundamental constants, with applications from astrophysics to laboratory spectroscopy experiments. We also provide a dedicated tool for the construction of hydrogenic transition amplitudes: "Computation of hydrogen radial INtegrals and COefficients" ($\texttt{CINCO}$).

hep-ph↗

arXiv2024

Axion Bounds from Quantum Technology

A consistent treatment of the quantum field theory of an axion-like particle (ALP) interacting with Standard Model fields requires to account for renormalisation group running and matching to the low-energy theory. Quantum sensor experiments designed to search for very light ALPs are particularly sensitive to these effects because they probe large values of the decay constant for which running effects become important. In addition, while linear axion interactions are set by its pseudoscalar nature, quadratic interactions are indistinguishable from scalar interactions. We show how the Wilson coefficients of linear and quadratic ALP interactions are related, including running effects above and below the QCD scale and provide a comprehensive analysis of the sensitivity of current and future experiments. We identify the reach of different experiments for the case of ALP dark matter and comment on how it could be distinguished from the case where it is not the dark matter. We present novel search strategies to observe quadratic ALP interactions via fifth force searches, haloscopes, helioscopes and quantum sensors. We emphasize the nonlinear behaviour of the ALP field close to the surface of the earth and point out which experimental results are independent on the local background field value.

hep-ph↗

arXiv2024

Liouville comparison theory for blowup of Euler-Arnold equations

In this article we introduce a new blowup criterion for (generalized) Euler-Arnold equations on $\mathbb R^n$. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and comparison with the Liouville equation; in contrast to the usual comparison approach at a single point, we apply comparison in an infinite dimensional function space. We thereby show that the Jacobian of the Lagrangian flow map of the solution reaches zero in finite time, which corresponds to $C^1$-blowup of the velocity field solution. We demonstrate the applicability of our result by proving blowup of smooth solutions to some higher-order versions of the EPDiff equation in all dimensions $n\geq 3$. Previous results on blowup of higher dimensional EPDiff equations were only for versions where the geometric description corresponds to a Sobolev metric of order zero or one. In these situations the behavior does not depend on the dimension and thus already solutions to the one-dimensional version were exhibiting blowup. In the present paper blowup is proved even in situations where the one-dimensional equation has global solutions, such as the EPDiff equation corresponding to a Sobolev metric of order two.

math.AP↗

arXiv2024

Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves

We investigate the geometry of the space of immersed closed curves equipped with reparametrization-invariant Riemannian metrics; the metrics we consider are Sobolev metrics of possible fractional order $q\in [0,\infty)$. We establish the critical Sobolev index on the metric for several key geometric properties. Our first main result shows that the Riemannian metric induces a metric space structure if and only if $q>1/2$. Our second main result shows that the metric is geodesically-complete (i.e., the geodesic equation is globally well-posed) if $q>3/2$, whereas if $q<3/2$ then finite-time blowup may occur. The geodesic-completeness for $q>3/2$ is obtained by proving metric-completeness of the space of $H^q$-immersed curves with the distance induced by the Riemannian metric.

math.DG↗

arXiv2024

Flavour bounds on axions, hidden photons and sterile neutrinos

Flavour-violating decays are some of the most sensitive probes for New Physics with masses below the B meson threshold. In this talk I review the sensitivity to axion-like particles, hidden photons and heavy neutral leptons assuming minimal models where these are the only particle in addition to the Standard Model, respectively.

hep-ph↗

arXiv2024

Sobolev metrics on spaces of manifold valued curves

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

math.DG↗

arXiv2024

How to measure the spin of invisible states in $e^+e^- \to γ+ X$

We examine the production of an invisible state $X$ together with a photon, $e^+e^- \to γ+X$, at electron positron colliders and present measurement strategies that can detect the spin of the invisible state as well as the underlying production mechanism, based on the angular distribution of the final state photon, the cross sections for polarized initial states and the photon polarization. Our measurement strategy can be used to identify whether the invisible state is a hidden photon or an axion. The results are compared with a detailed analysis of the Standard Model background and we calculate the sensitivity reach for searches for axions and hidden photons at Belle II.

hep-ph↗

arXiv2023

Geometric Analysis of the Generalized Surface Quasi-Geostrophic Equations

We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach.

math.DG↗

arXiv2023

Basis restricted elastic shape analysis on the space of unregistered surfaces

This paper introduces a new mathematical and numerical framework for surface analysis derived from the general setting of elastic Riemannian metrics on shape spaces. Traditionally, those metrics are defined over the infinite dimensional manifold of immersed surfaces and satisfy specific invariance properties enabling the comparison of surfaces modulo shape preserving transformations such as reparametrizations. The specificity of the approach we develop is to restrict the space of allowable transformations to predefined finite dimensional bases of deformation fields. These are estimated in a data-driven way so as to emulate specific types of surface transformations observed in a training set. The use of such bases allows to simplify the representation of the corresponding shape space to a finite dimensional latent space. However, in sharp contrast with methods involving e.g. mesh autoencoders, the latent space is here equipped with a non-Euclidean Riemannian metric precisely inherited from the family of aforementioned elastic metrics. We demonstrate how this basis restricted model can be then effectively implemented to perform a variety of tasks on surface meshes which, importantly, does not assume these to be pre-registered (i.e. with given point correspondences) or to even have a consistent mesh structure. We specifically validate our approach on human body shape and pose data as well as human face scans, and show how it generally outperforms state-of-the-art methods on problems such as shape registration, interpolation, motion transfer or random pose generation.

cs.CV↗

arXiv2023

Isometric Immersions and the Waving of Flags

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values in $\mathbb R^3$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.

math.DG↗

arXiv2023

Square Root Normal Fields for Lipschitz surfaces and the Wasserstein Fisher Rao metric

The Square Root Normal Field (SRNF) framework is a method in the area of shape analysis that defines a (pseudo) distance between unparametrized surfaces. For piecewise linear (PL) surfaces it was recently proved that the SRNF distance between unparametrized surfaces is equivalent to the Wasserstein Fisher Rao (WFR) metric on the space of finitely supported measures on $S^2$. In the present article we extend this point of view to a much larger set of surfaces; we show that the SRNF distance on the space of Lipschitz surfaces is equivalent to the WFR distance between Borel measures on $S^2$. For the space of spherical surfaces this result directly allows us to characterize the non-injectivity and the (closure of the) image of the SRNF transform. In the last part of the paper we further generalize this result by showing that the WFR metric for general measure spaces can be interpreted as an optimization problem over the diffeomorphism group of an independent background space.

math.DG↗

arXiv2023

BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes

We present Basis Restricted Elastic Shape Analysis (BaRe-ESA), a novel Riemannian framework for human body scan representation, interpolation and extrapolation. BaRe-ESA operates directly on unregistered meshes, i.e., without the need to establish prior point to point correspondences or to assume a consistent mesh structure. Our method relies on a latent space representation, which is equipped with a Riemannian (non-Euclidean) metric associated to an invariant higher-order metric on the space of surfaces. Experimental results on the FAUST and DFAUST datasets show that BaRe-ESA brings significant improvements with respect to previous solutions in terms of shape registration, interpolation and extrapolation. The efficiency and strength of our model is further demonstrated in applications such as motion transfer and random generation of body shape and pose.

cs.CV↗

arXiv2023

The $L^p$-Fisher-Rao metric and Amari-Cencov $α$-connections

We introduce a family of Finsler metrics, called the $L^p$-Fisher-Rao metrics $F_p$, for $p\in (1,\infty)$, which generalizes the classical Fisher-Rao metric $F_2$, both on the space of densities Dens$_+(M)$ and probability densities Prob$(M)$. We then study their relations to the Amari-uCencov $α$-connections $\nabla^{(α)}$ from information geometry: on Dens$_+(M)$, the geodesic equations of $F_p$ and $\nabla^{(α)}$ coincide, for $p = 2/(1-α)$. Both are pullbacks of canonical constructions on $L^p(M)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of $α$-geodesics as being energy minimizing curves. On Prob$(M)$, the $F_p$ and $\nabla^{(α)}$ geodesics can still be thought as pullbacks of natural operations on the unit sphere in $L^p(M)$, but in this case they no longer coincide unless $p=2$. Using this transformation, we solve the geodesic equation of the $α$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman-Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of $F_p$, and study their relation to $\nabla^{(α)}$.

math.DG↗